This work results from our attempts to solve Boltzmann-Sinai's hypothesis about the ergodicity of hard ball gases. A crucial element in the studies of the dynamics of hard balls is the analysis of special hypersurfaces in the phase space consisting of degenerate trajectories (which lack complete hyperbolicity). We prove that if a flow-invariant hypersurface $J$ in the phase space of a semi-dispersing billiard has a negative Lyapunov function, then the volume of the forward image of $J$ grows at least linearly in time. Our proof is independent of the solution of the Boltzmann-Sinai hypothesis, and we provide a complete and self-contained argument here.
In this paper we study the singularity manifolds of multidimensional strictly dispersing billiards and show that the proof of the Fundamental theorem for dispersing billiards remain valid for a dense set of finitely smooth scatterers.
We consider dispersing billiard tables whose boundary is piecewise smooth and the free flight function is unbounded. We also assume there are no cusps. Such billiard tables are called type D in the monograph of Chernov and Markarian. For a class of non-degenerate type D dispersing billiards, we prove exponential decay of correlation and several other statistical properties.
It is known that the dynamics of planar billiards satisfies strong mixing properties (e.g. exponential decay of correlations) provided that some expansion condition on unstable curves is satisfied. This condition has been shown to always hold for smooth dispersing planar billiards, but it needed to be assumed separately in the case of dispersing planar billiards with corner points. We prove that this expansion condition holds for any dispersing planar billiard with corner points, no cusps and bounded horizon.
Local finiteness of the number of reflections in semi-dispersing Minkowski billiards
St. Petersburg Department of V. A. Steklov Mathematical Institute of the Russian Academy of Sciences（俄罗斯科学院V.A.斯捷克洛夫数学研究所圣彼得堡分部）；St. Petersburg State University（圣彼得堡国立大学）
We prove exponential correlation decay in dispersing billiard flows on the 2-torus assuming finite horizon and lack of corner points. With applications aimed at describing heat conduction, the highly singular initial measures are concentrated here on 1-dimensional submanifolds (given by standard pairs) and the observables are supposed to satisfy a generalized Hölder continuity property. The result is based on the exponential correlation decay bound of Baladi, Demers and Liverani obtained recentlyfor Hölder continuous observables in these billiards. The model dependence of the bounds is also di…
Dynamical billiards, or the behavior of a particle traveling in a planar region $D$ undergoing elastic collisions with the boundary, has been extensively studied and is used to model many physical phenomena such as a Boltzmann gas. Of particular interest are the dispersing billiards, where $D$ consists of a union of finitely many open convex regions. These billiard flows are known to be ergodic and to possess the $K$-property. However, Turaev and Rom-Kedar (1998) proved that for dispersing systems permitting singular periodic orbits, there exists a family of smooth Hamiltonian flows with regio…
We consider the class of dispersing billiard systems in the plane formed by removing three convex analytic scatterers satisfying the non-eclipse condition. The collision map in this system is conjugated to a subshift, providing a natural labeling of periodic points. We study the problem of marked length spectrum rigidity for this class of systems. We show that two such systems have the same marked length spectrum if and only if their collision maps are analytically conjugate to each other near a homoclinic orbit and that two scatterers and the marked length spectrum together uniquely determine…
We consider billiards obtained by removing three strictly convex obstacles satisfying the non-eclipse condition on the plane. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift on three symbols that provides a natural labeling of all periodic orbits. We study the following inverse problem: does the Marked Length Spectrum (i.e., the set of lengths of periodic orbits together with their labeling), determine the geometry of the billiard table? We show that from the Marked Length Spectrum it is possible to recover the curvature at periodic points of perio…
Since the seminal work of Sinai one studies chaotic properties of planar billiards tables. Among them is the study of decay of correlations for these tables. There are examples in the literature of tables with exponential and even polynomial decay. However, until now nothing is known about mixing properties for billiard tables with non-compact cusps. There is no consensual definition of mixing for systems with infinite invariant measure. In this paper we study geometric and ergodic properties of billiard tables with a non-compact cusp. The goal of this text is, using the definition of mixing p…
Given an integer $k \geq 5$, and a $C^k$ Anosov flow $Φ$ on some compact connected $3$-manifold preserving a smooth volume, we show that the measure of maximal entropy (MME) is the volume measure if and only if $Φ$ is $C^{k-\varepsilon}$-conjugate to an algebraic flow, for $\varepsilon>0$ arbitrarily small. Besides the rigidity, we also study the entropy flexibility, and show that the metric entropy with respect to the volume measure and the topological entropy of suspension flows over Anosov diffeomorphisms on the $2$-torus achieve all possible values subject to natural normalizations. Moreov…
In this paper we constructed a special family of semidispersing billiards bounded on a rectangle with a few dispersing scatters. We assume there exists a pair of flat points (with zero curvature) on the boundary of these scatters, whose tangent lines form a channel in the billiard table that is perpendicular to the vertical sides of the rectangle. The billiard can be induced to a Lorenz gas with infinite horizon when replacing the rectangle by a torus. We study the mixing rates of the one-parameter family of the semi-dispersing billiards and the Lorenz gas on a torus; and show that the correla…
Following recent work of Chernov, Markarian, and Zhang, it is known that the billiard map for dispersing billiards with zero angle cusps has slow decay of correlations with rate 1/n. Since the collisions inside a cusp occur in quick succession, it is reasonable to expect a much faster decay rate in continuous time. In this paper we prove that the flow is rapid mixing: correlations decay faster than any polynomial rate. A consequence is that the flow admits strong statistical properties such as the almost sure invariance principle, even though the billiard map does not. The techniques in this p…
Let $f: [0, +\infty) \to (0, +\infty)$ be a sufficiently smooth convex function, vanishing at infinity. Consider the planar domain $Q$ delimited by the positive $x$-semiaxis, the positive $y$-semiaxis, and the graph of $f$. Under certain conditions on $f$, we prove that the billiard flow in $Q$ has a hyperbolic structure and, for some examples, that it is also ergodic. This is done using the cross section corresponding to collisions with the dispersing part of the boundary. The relevant invariant measure for this Poincaré section is infinite, whence the need to surpass the existing results, de…
In this short note we consider the finite-dimensional distributions of sets of states generated by dispersing billiards with a random initial condition. We establish a functional correlation bound on the distance between the finite-dimensional distributions and corresponding product distributions. We demonstrate the usefulness of the bound by showing that it implies several limit theorems. The purpose of this note is to provide a tool facilitating the study of more general functionals of the billiard process.
We propose a model of Sinai billiards with moving scatterers, in which the locations and shapes of the scatterers may change by small amounts between collisions. Our main result is the exponential loss of memory of initial data at uniform rates, and our proof consists of a coupling argument for non-stationary compositions of maps similar to classical billiard maps. This can be seen as a prototypical result on the statistical properties of time-dependent dynamical systems.
The Local Ergodic Theorem (also known as the `Fundamental Theorem') gives sufficient conditions under which a phase point has an open neighborhood that belongs (mod 0) to one ergodic component. This theorem is a key ingredient of many proofs of ergodicity for billiards and, more generally, for smooth hyperbolic maps with singularities. However the proof of that theorem relies upon a delicate assumption (Chernov-Sinai Ansatz), which is difficult to check for some physically relevant models, including gases of hard balls. Here we give a proof of the Local Ergodic Theorem for two dimensional bill…
We consider a class of planar dispersing billiards with a cusp at a point of vanishing curvature. Convergence to a stable law and to the corresponding Lévy process in the $\cM_1$ and $\cM_2$ Skorohod topologies has been studied in recent work. Here we show that certain sufficient conditions for $\cM_2$-convergence are also necessary.
For any finite horizon Sinai billiard map T on the two-torus, we find t_*>1 such that for each t in (0,t_*) there exists a unique equilibrium state $μ_t$ for $- t\log J^uT$, and $μ_t$ is T-adapted. (In particular, the SRB measure is the unique equilibrium state for $- \log J^uT$.) We show that $μ_t$ is exponentially mixing for Holder observables, and the pressure function $P(t)=\sup_μ\{h_μ-\int t\log J^uT d μ\}$ is analytic on (0,t_*). In addition, P(t) is strictly convex if and only if $\log J^uT$ is not $μ_t$ a.e. cohomologous to a constant, while, if there exist $t_a\ne t_b$ with $μ_{t_a}= …
We consider a multi-dimensional billiard system in an (n+1)-dimensional Euclidean space, the direct product of the "horizontal" hyperplane and the "vertical" line. The hypersurface that determines the system is assumed to be smooth and symmetric in all coordinate hyperplanes. Hence there exists a periodic orbit $γ$ of period 2 moving along the "vertical" coordinate axis. The question we ask is as follows. Is it possible to choose such a system to have the dynamics locally (near $γ$) conjugated to the dynamics of a linear map? Since the problem is local, the billiard hypersurface can be determi…
We describe a one-parameter family of dispersing (hence hyperbolic, ergodic and mixing) billiards where the correlation function of the collision map decays as $1/n^a$ (here $n$ denotes the discrete time), in which the degree $a \in (1, \infty)$ changes continuously with the parameter of the family, $β$. We also derive an explicit relation between the degree $a$ and the family parameter $β$.
For Dynamical Systems, a strong bound on multiple correlations implies the Central Limit Theorem (CLT) [ChMa]. In Chernov's paper [Ch2], such a bound is derived for dynamically Holder continuous observables of dispersing Billiards. Here we weaken the regularity assumption and subsequently show that the bound on multiple correlations follows directly from the bound on pair correlations. Thus, a strong bound on pair correlations alone implies the CLT, for a wider class of observables. The result is extended to Anosov diffeomorphisms in any dimension.
In this note we study the higher dimensional convex billiards satisfying the so-called Gutkin property. A convex hypersurface $S$ satisfies this property if any chord $[p,q]$ which forms angle $δ$ with the tangent hyperplane at $p$ has the same angle $δ$ with the tangent hyperplane at $q$. Our main result is that the only convex hypersurface with this property in $\mathbf{R}^d, d\geq 3$ is a round sphere. This extends previous results on Gutkin billiards obtained in \cite{B0}.
We show a rigidity result for 3-dimensional contact Axiom A flows: given two 3D contact Axiom A flows $Φ_1,Φ_2$ whose restrictions $Φ_1|_{Λ_1},Φ_2|_{Λ_2}$ to basic sets $Λ_1,Λ_2$ are orbit equivalent, we prove that if periodic orbits in correspondence have the same length, then the conjugacy is as regular as the flows and respects the contact structure, extending a previous result due to Feldman-Ornstein [21]. Some of the ideas are reminiscent of the work of Otal [51]. As an application, we show that the billiard maps of two open dispersing billiards without eclipse and with the same marked le…
For 2-d hyperbolic systems with singularities, statistical properties are rather difficult to establish because of the fragmentation of the phase space by singular curves. In this paper, we construct a Markov partition of the phase space with countable states for a general class of hyperbolic systems with singularities. Stochastic properties with respect to the SRB measure immediately follow from our construction of the Markov partition, including the decay rates of correlations and the central limit theorem. We further establish the thermodynamic formalism for the family of geometric potentia…
In the ergodic theory of semi-dispersing billiards the Local Ergodic Theorem, proved by Chernov and Sinai in 1987, plays a central role. So far, all existing proofs of this theorem had to use an annoying global hypothesis, namely the almost sure hyperbolicity of singular orbits. (This is the so called Chernov--Sinai Ansatz.) Here we introduce some new geometric ideas to overcome this difficulty and liberate the proof from the tyranny of the Ansatz. The presented proof is a substantial generalization of my previous joint result with N. Chernov (which is a $2D$ result) to arbitrary dimensions. A…
Let $v$ be a unit vector field on a complete, umbilic (but not totally geodesic) hypersurface $N$ in a space form; for example on the unit sphere $S^{2k-1} \subset \mathbb{R}^{2k}$, or on a horosphere in hyperbolic space. We give necessary and sufficient conditions on $v$ for the rays with initial velocities $v$ (and $-v$) to foliate the exterior $U$ of $N$. We find and explore relationships among these vector fields, geodesic vector fields, and contact structures on $N$. When the rays corresponding to each of $\pm v$ foliate $U$, $v$ induces an outer billiard map whose billiard table is $U$. …
The configuration manifold $M$ of a mechanical system consisting of two unconstrained rigid bodies in $\mathbb{R}^n$, $n\geq 1$, is a manifold with boundary (typically with singularities.) A complete description of the system requires boundary conditions that specify how orbits should be continued after collisions. A boundary condition is the assignment of a collision map at each tangent space on the boundary of $M$ that gives the post-collision state of the system as a function of the pre-collision state. Our main result is a complete description of the space of linear collision maps satisfyi…
